Utility functions for financial calculators Anthony Asher and Gaurav Khemka June 2015 (with co-authors Adam Butt and Ujwal Kayande)

Transcription

1 Utility functions for financial calculators Anthony Asher and Gaurav Khemka June 2015 (with co-authors Adam Butt and Ujwal Kayande) This research was supported by the Centre for International Finance and Regulation (project number T024), which is a Centre of Excellence for research and education in the financial sector, funded by the Commonwealth and NSW Governments and supported by other consortium members (see

2 Introduction People need help to plan for retirement. Some of the help must come in the form of a calculator to decide how much to save and to spend, and where to invest. These are the decisions that people can control (in the cab). Current advice and calculators are embarrassingly primitive and much panders to ignorance and instinct (some fortunately is not wrong!) Users do not need to know the details of the workings of the calculators, but those who design them need to use plausible and coherent utility functions that represent the users preferences and relevant demographic and economic assumptions. These functions can be as complex as necessary (because they belong under the hood). Users ultimately have to express preferences between alternative stochastic distributions of random vectors of consumption, leisure and bequests. This is not easy but can perhaps be elicited through offering them graphic representations. 2

3 Current advice is primitive Many are financially illiterate and take no interest in retirement savings. Advice can be inconsistent, biased and counter-productive (Gokhale and Kotlikoff; Inderst and Ottaviani; Hackethal et al). Merton (2014) suggests that we face a crisis in financial well-being. Inappropriate inflation and investment assumptions equity risk premium too high; no momentum and mean reversion. No allowance for consistency between current and future consumption by using arbitrary targets or needs. Misleading risk appetite assessments using short term portfolio volatility and life expectancy. Incoherent ruin probabilities (VaR), which can encourage risk taking by ignoring the size of some downside losses. 3

4 From Moneysmart, ASIC sponsored 4

5 From Moneysmart, ASIC sponsored As my wife and I are earning $154,000 pa we are assumed to have an income goal of $100,000 pa and are advised to contribute $67,000 pa to make up the shortfall. She has a 50:50 chance of living past ninety, but the money runs out before then - pity! 5

6 Identifying and addressing ignorance and instinct Bounded rationality, bounded willpower The difficulties with traditional logic can be explained by two systems : System 1 is instinctive, responds rapidly, is low effort, often quick to make emotional judgments and works off rules of thumb System 2 is slower, reasoned and requires higher effort to engage, although not always right. The difficulty of engaging system 2 means that we all have a tendency to make snap judgements that we can regret, because we did not take the time and effort to engage our deeper thinking processes. But the lessons of modern psychology (Mischel, 2014) and the ancient world religions tell us we can with some effort decide differently. So then with the mind I myself serve the law of God; but with the flesh I serve the law of sin. (Romans 7:25 King James Version) It is surely possible to provide training so that people can plan better more consistently and in a way where they implement their decisions. 6

7 Explaining apparent irrationality System 1 heuristics explain some apparent irrationality: Hyperbolic discounting, with high short term rates of discount for small amounts of money. Discount rates are lower when people consider probabilities (too complex for system 1). Similarly, when time preferences are framed as choosing lifetime patterns of consumption, people prefer level ones. People choose prior commitments (such as forced savings and bucket allocations) to take decisions away from system 1. Other apparent irrationality can be explained by frictional costs: The Prospect Theory kink in the utility function can be explained by consequential financial losses (eg cannot use the car and so sell it). Information costs of making equity investments mean greater risk aversion. If my consumption falls below those of my reference group, I cannot participate as much in their activities. (See Frederick et al (2002) and Clark et al (2000) for explorations of some of these biases but without these explanations.) Also need to take into account in reserve perhaps hedonic adjustment means that failing to maintain consumption is not a disaster, particularly when a basic state support is provided. 7

8 The standard utility function UU 0 = TT 1 ββkk tt 1 ( kk 2 1 tt=0 pp jj ){pp tt 1 1 γγ CC tt ll αα 1 γγ + bb(1 pp tt 1 ) 1 DD tt 1 γγ tt 1 γγ } Where β represents the net time preference rate; C t consumption in period t; φ t represents an allowance for the family state (which in Hubener et al (2013) is contingently modelled using empirical transition rates for divorce and widowhood; l α represents the utility of the hours of leisure; p t the probability of survival in the year t to t+1; γ>0 is the coefficient of risk aversion; b the intensity of the bequest motive, and D t the bequest. 8

9 Possible adaptations UU 0 = TT 1 ββ tt 1 ( tt 2 1 jj=0 pp jj ){pp tt 1 1 γγ CC tt ll αα 1 γγ + bb(1 pp tt 1 ) 1 DD tt 1 γγ tt 1 γγ } β should perhaps be contingent on actual investment returns because of reference group to represent level consumption throughout life. Φ t can be partly controlled; should one include probabilities of divorce? l α could include a constraint related to age and health in order to generate a retirement date, or retirement date could be an input variable. p t could also be set to 1 as any coherent plan that has zero utility for the state of death would use a life annuity γ>0 can perhaps be enhanced/replaced by a penalty whenever consumption drops, which would also produce smoothing. Alternatively consider intertemporal elasticity; constant IEC implies (essentially): UU 0 = (cc 0 +cc1 ) 1 ρρ b and D t can be removed, and an amount set aside separately for precautionary ρρ ρρ and bequest purposes. 9

10 10 Eliciting the utility function

11 An Example: Utility Specification The modified version of the CRRA utility function described earlier is used. TT UU 0 = tt=1 In this formulation: tt 1 pp jj CC tt 1 γγ jj=0 1 γγ 1 γγ 1 γγ + ξξ min 0, CC tt 1 γγ CC tt 1 1 γγ A penalty ξξ = 0.1 applies where the current period s consumption is lower than the past period s consumption Bequest motives are not allowed for Leisure or family states are not considered Appropriate timing adjustments are not shown pp tt The optimisation procedure involves a stochastic dynamic programming process 11

12 An Example: Illustration of a simple process (no penalty) For each simulation at age x: VV XX = vv xx + VV(xx + 1) 1 qq xx Age x Age x+1 V(x) Starting Balance v(x) Ending Balance Bal1 Bal2 Bal3 Obj1 Obj2 Obj3 RAA, C/W Bal20 Bal21 Obj20 Obj21 12

13 An Example: Input and Output Inputs and Assumptions Economic data for simulation bootstrapped from relevant indices over the period results expressed on a real (AWE) basis Mortality based on male ALT rates no mortality improvement allowed for Ages considered are 25 through to 109 with retirement at 65. Labour is the only source of income pre-retirement and superannuation is the only source of income post-retirement Outputs Optimum results (not shown here) are generated for a combination of balances and prior consumption values (state variables). The resulting output provides optimum risky asset allocation and consumption amounts. 13

14 14 An Example: Median Outcomes for age 25

15 15 An Example: Median Outcomes for age 65

16 16 An Example: Median Path for age 25

17 17 An Example: Median Path for age 65

18 An Example: Comparison Comparison: Age 25 Balance $ Cons (no penalty) Cons (penalty) Balance (no enalty) Balance (Penalty) cons25 cons27 cons29 cons31 cons33 cons35 cons37 cons39 cons41 cons43 cons45 cons47 cons49 cons51 cons53 cons55 cons57 cons59 cons61 cons63 cons65 cons67 cons69 cons71 cons73 cons75 cons77 cons79 cons81 cons83 cons85 cons87 cons89 cons91 cons93 cons95 cons97 cons99 cons101 cons103 cons105 cons107 cons109 0

19 Future Steps Experiment with alternative ways of eliciting utility functions or is coherence an illusion and different preferences incommensurable? Develop realistic calculator o Australian social security and taxation o Marital status, children, home ownership etc o Add alternative annuities Develop alternative presentations to increase engagement Add further asset classes and refine optimisation procedures 19